Exercise 17(vi) Log plot

Gary

In (31) it says that "...$\widetilde{Q}$ rotates most rapidly and its size remains constant, when $Q$ moves in the direction of $\mathcal{K}$...".

This means that as we travel on the LHS, from the white circle at 150 degrees to the opposite white circle at -30 degreees, along the diameter in the direction of $\mathcal{K}$, the lever on the image circles, (traversed in the direction north, centre, south) on the RHS, rotate anticlockwise through a larger angle than for any other diameter on the LHS. It seems to me that figure 2 is in agreement with this. It may not be correct overall but it does agree with (31).

I have calculated the total amount of rotation for the six diameters and the maximum value is indeed along the north-south diameter.

Similarly, in the direction of $-i\mathcal{K}$ on the LHS, the rotation on the RHS is zero and the amplification on the RHS is at its greatest, which again agrees with (31).

A good way to think about (21) in my opinion, is as a shape at some point in the complex plane, with its image under $f$ in the image complex plane. We can now move the original point in any direction from its original position and watch what happens to its image. In one of these directions, $\mathcal{K}$, the image rotates most rapidly, without changing shape, and in the direction $-i\mathcal{K}$, at right angles to the direction of $\mathcal{K}$, the opposite is true, and there is no rotation but maximum magnification. In all the other directions both rotation and magnification change. As we move from the direction $\mathcal{K}$ towards the direction $-i\mathcal{K}$, the image rotation along the diameter decreases and the magnification along the diameter increases until we reach the direction $-i\mathcal{K}$, and if we keep going the reverse happens.

We are interested here in movement along the spokes of the wheel, not around the circumference.

Vasco

Vasco,

Here is my version of the Log plot, finally. I anticipate that the remainder of 17 will go more quickly. I see an advantage in your method of using the spreadsheet for calculations, as you have the ability to monitor the numbers. I had trouble plotting this until I obtained more succinct expressions for $\alpha$ and $\sigma$.

nh.ch5.ex17.vi.log.pdf (117.86 KB)

I see that the .pdf has a faint plot. I tried resaving and reloading it, but to no avail. I'll see what I can do. Meanwhile, I think the essentials can be read from it. Here, at least, is a .jpg file.



Gary

Midnight oil, yes, and for just one part of many of one problem in a text with many problems. No one can accuse Needham of making life easy for students. Thank you for the comments. But now that I have some idea of a valid approach, I'm looking forward to trying the z^m and Möbius problems. I look forward to your final document, but perhaps you should warn the readers that this is not an ordinary problem to be dispatched in a page or two! Thank you for the comments. I will amend the figure.

Gary